6 research outputs found

    Type-Two Well-Ordering Principles, Admissible Sets, and Pi^1_1-Comprehension

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    This thesis introduces a well-ordering principle of type two, which we call the Bachmann-Howard principle. The main result states that the Bachmann-Howard principle is equivalent to the existence of admissible sets and thus to Pi^1_1-comprehension. This solves a conjecture of Rathjen and Montalbán. The equivalence is interesting because it relates "concrete" notions from ordinal analysis to "abstract" notions from reverse mathematics and set theory. A type-one well-ordering principle is a map T which transforms each well-order X into another well-order T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our Bachmann-Howard principle transforms each dilator T into a well-order BH(T). The latter is a certain kind of fixed-point: It comes with an "almost" monotone collapse theta:T[BH(T)]->BH(T) (we cannot expect full monotonicity, since the order-type of T[X] may always exceed the order-type of X). The Bachmann-Howard principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is "computed". On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for set-theoretic reverse mathematics; a formalization of dilators in weak set theories and second-order arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (Schütte) and beta-completeness (Girard) in a set-theoretic context; and a beta-consistency proof for Kripke-Platek set theory. Independently of the Bachmann-Howard principle, the thesis contains a series of results connected to slow consistency (introduced by S.-D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the Paris-Harrington principle can only have extremely long proofs in certain fragments of arithmetic

    Well ordering principles for iterated 511-comprehension

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    We introduce ordinal collapsing principles that are inspired by proof theory but have a set theoretic flavor. These principles are shown to be equivalent to iterated 1 1- comprehension and the existence of admissible sets, over weak base theories. Our work extends a previous result on the non-iterated case, which had been conjectured in Montalbán’s “Open questions in reverse mathematics" (Bull Symb Log 17(3):431– 454, 2011). This previous result has already been applied to the reverse mathematics of combinatorial and set theoretic principles. The present paper is a significant contribution to a general approach that connects these fields

    Applicazioni di analisi ordinale: il teorema di Kruskal.

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    Si fornisce una caratterizzazione del teorema di Kruskal con la tecnica dell'analisi ordinale, uno dei principali strumenti della teoria della dimostrazione

    Two applications of analytic functors

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    AbstractWe apply the theory of analytic functors to two topics related to theoretical computer science. One is a mathematical foundation of certain syntactic well-quasi-orders and well-orders appearing in graph theory, the theory of term rewriting systems, and proof theory. The other is a new verification of the Lagrange–Good inversion formula using several ideas appearing in semantics of lambda calculi, especially the relation between categorical traces and fixpoint operators
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